Can I get legal residency after my J1 visa expired?

If you are still in the US on the J-1, then your I-94 card will say "d/s" on it. As long as DHS has not terminated your status in the meantime, when you leave the US, you will not be subject to the three or ten year bar from reentering the US. So technically you would be able to reapply for entry. However the US consulate will probably not give you a visa, because you remained in the US when you should not have. So my best recommendation is to re-establish yourself abroad for two years- employment, lease, etc and then to try to re-enter at that time. Your current status (unless DHS has terminated it) is that you are in J-1 status but you are in violation of that status. So you cannot apply to change status to any other non-immigrant visa in the US.